algorithm orals 2002 1 algorithm qualifying examination orals achieving 100% throughput in iq/cioq...

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Algorithm Orals 2002

High PerformanceSwitching and RoutingTelecom Center Workshop: Sept 4, 1997.

Algorithm Qualifying Examination Orals

Achieving 100% Throughput in IQ/CIOQ Switches using Maximum Size and Maximal Matching Algorithms

Sundar IyerStanford University

[email protected]/~sundaes


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Outline

Introduction

Part-I: Properties of Maximum Size Matching (MSM) in an IQ switch

Stability of critical MSM for any Bernoulli i.i.d. traffic Stability of MSM for Bernoulli i.i.d. uniform traffic

Part-II: Properties of Maximal Matching (MXM) in a CIOQ switch

A simple proof for stability


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Simple Model of a Switch

Port 1, input Port 1, output




R

R

R

R

R

R

R

R

Example: Output Queued Switch


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Input Queued Switch Model

N N

1 1R

R

Example: Input Queued Switch with virtual output queues (VOQs)

Crossbar

R

R

Port 1, input

Port N, input

Port 1, output

Port 4, output

VOQs


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Relation to a Graph Matching

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VOQs


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Classes of Scheduling Algorithms

Maximum Weight Matching (MWM)

Choose a matching which maximizes the weight of the matching

MWM gives 100% throughput

Maximum Size Matching (MSM)

Choose a matching which maximizes the size of the matching


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Outline

Introduction






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MSM is Unstable

N N

1 1

Request Graph

N N

1 1

N N

1 1

..

N N

1 1

Switch schedule based on MSM

T=1 T=2 ……….


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Questions

Are all MSMs unstable?

Is there a subclass of MSMs which are stable? There is at least one MSM which is stable.

Are MSMs stable under uniform load?

Simulation seems to suggest this. Can we prove this?


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Non Pre-emptive SchedulingBatch Scheduling

N N

1 1R

R

Priority-2

Crossbar

R

R

Port 1, input

Port N, input

Port 1, output

Port N, output

Priority-1

Batch-(k+1)

Batch-(k)


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Non Pre-emptive SchedulingBatch Scheduling

N N

1 1R

R

Priority-2

Crossbar

R

R

Port 1, input

Port N, input

Port 1, output

Port N, output

Priority-1

Batch-(k+1)

Batch-(k)


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Degree of a Batch

1

2

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0

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0

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1

0

0

0

1

1

2

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Batch Request GraphDegree (dv,k):

The number of cells departing from (destined to) a vertex in batch k.

Maximum Degree (Dk) The maximum degree

amongst all inputs/outputs in batch k.


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Critical Maximum Size Matching

2

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12

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Batch Request Graph

degree =3 degree =3


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Outline

Introduction


Stability of Critical MSM for any Bernoulli i.i.d. traffic Stability of MSM for Bernoulli i.i.d. uniform traffic


A Simple proof for stability


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The Arrival Process

, :

1, 1

ij ij

ij iji j

A

1. Traffi c matrix:

where expected number of

arrivals in one timeslot

2. I f ; we say the traffi c is "admissible".

3. For a Bernoulli i.

, ( , );ij i jN

i.d arrival process:

I f we say the traffi c is unif orm.


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Stability of CMSM

Theorem 1:

CMSM is stable under batch scheduling, if the input traffic is admissible and Bernoulli i.i.d. uniform

Informal Arguments: Let Tk be the time to schedule batch k

Then for batch k+1 we buffer packets for time Tk

We expect about Tk packets at every input/output

Hence, the maximum degree of batch k +1, i.e. Dk+1 Tk

Hence for a CMSM Tk+1 = Dk+1 = Tk < Tk

Hence Tk converges to a finite number


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Formal Arguments … 1 We shall use the Chernoff bound to get

If we want to bound Dk, we require that all the 2N vertices are bounded

(1 ), 1{ (1 ) }

(1 )

kT

v k k veP d T p

1{ (1 ) } 1 2k k vP D T Np Q


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We can choose (1 + ) < 1 - to get

Observe that Q is now a function of Tk only.

We can make Q as close to 1, by choosing a large Tk

Also, Tk+1 NTk

This gives

Formal Arguments … 2

1{ (1 ) }k kP T T Q

1( ) (1- ) (1- )

(1- ) , if .....

k k

k

kE T Q T Q NT

T


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Formal Arguments …3

Hence, there is a constant Tc which depends only on (and hence only on ), such that

Formally, using a linear Lyapunov function V(Tk) = Tk, we can say that E(Tk) is bounded.

1( ) (1- ) , k k k cE T T T T


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Stability of CMSM

Theorem 2:

CMSM is stable under batch scheduling, if the input traffic is admissible and Bernoulli i.i.d.


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Outline

Introduction


Stability of Critical MSM for any Bernoulli i.i.d. traffic Stability of MSM for Bernoulli i.i.d. uniform traffic


A Simple proof for stability


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Example of a Uniform Graph

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Batch Request Graph

degree =3 degree =3


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Properties of Uniform Graphs

Lemma-1:

If the request graph is uniform and the maximum degree is D, then any MSM can schedule the requests in exactly D time slots

Lemma-2:

Any request graph with maximum degree D, can be scheduled by any MSM within 2D time slots


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Property of any Graph Theorem:

Any request graph with maximum degree is D, and minimum VOQ length m, can be scheduled in less than 2D –Nm time slots

Proof: Consider a request graph with minimum VOQ length m The minimum degree of the graph is mN Hence the original graph can be considered to be in two

parts

• A uniform graph of degree mN

• Another graph of maximum degree D – mN

Hence the request graph can be scheduled in at most mN + 2(D-mN) = 2D - Nm


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Stability of MSM ..1

Theorem 3:

MSM is stable under batch scheduling, if the input traffic is admissible and Bernoulli i.i.d. uniform

Informal Arguments

We can bound both the maximum degree D and the minimum VOQ length m

The rest of the proof is similar to the CMSM proof


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Outline

Introduction






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Maximal Matching Algorithms

Maximal Matching (MXM)

Choose a matching such that no unmatched input or output has a packet meant for each other

They are easier to implement and have low complexity

They are known to be unstable and give low throughput for input queued switches


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A Model for a CIOQ switch

Combined Input-Output Queued Switch

Bandwidth: 2NR

2R

2R

2R

Port 1

Port 2

Port N

2R

2R

2R

R

R

R

Port 1

Port 2

Port N

R

R

R

A CIOQ switch with a speedup of 2, gives 100% throughput for any MXM algorithm

• [Ref: Dai & Prabhakar, Leonardi. et. al.]


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Let Aj(t1,t2) denote the number of arrivals to output j in the interval between (t1,t2)

A leaky bucket constrained traffic satisfies, the property that for each output j

Note that this means that for an ideal output queued switch no output has more than B packets in the switch

Let DT denote the departure time of a packet from this ‘ideal’ output queued switch

Leaky Bucket Traffic

1 2 2 1 j( , ) ( ) ; where α <1j jA t t t t B


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Stability of MXM

Theorem 4:

A CIOQ switch with an MXM algorithm gives bounded delay and hence 100% throughput with a speedup greater than 2, under arrivals which satisfy the leaky bucket constraint


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Constraint Set ‘Maximal’ Algorithm

The algorithm is greedy i.e. when a cell arrives, it immediately attempts to allot a time (in the future) when it should be transferred

Each input and output maintains a constraint set of the future times during which it is free to send/receive a packet

The algorithm attempts to bound the time of departure of a packet to within k time slots of its departure time DT, i.e each packet is transferred in the time (DT, DT+k)


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Allocations as seen by the Output

…DT + k DT- kDT

ck

Packet has an OQ Departure Time = DT

Packet should leave in the interval (DT, DT + k)

In the interval (DT, DT + k) There is one cell which tries to get allotted in that interval. No more than k cells get delayed and are allotted to that interval

Number of Time Slots Available is more than k

kS


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Allocations as seen by the Input

…DT + k DT-B-kDT

B + k

DT-B

Packet has an OQ Departure Time = DT

Packet should leave during interval (DT, DT + k)

In the interval (DT, DT + k) There is one cell which tries to get allotted in that interval No cell which arrived before DT–B-k will be allotted to this interval

Number of Time Slots Available is more than

ck

k Bk

S


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Sufficiency Conditions on Speedup

We are guaranteed a timeslot if

The above equation can be satisfied if

This means S > 2 is sufficient to guarantee that the delay is bounded

This implies 100% throughput

k B kk k k

S S

2

Bk

S


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Stability of MXM

Theorem 5:

A CIOQ switch with an MXM algorithm gives 100%throughput with a speedup greater than 2, under

admissiblearrivals which satisfy the strong law of large numbers


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Summary

In an IQ switch with batch scheduling

A subclass of MSM called CMSM is stable, if the input traffic is admissible and Bernoulli i.i.d.

MSM is stable, if the input traffic is admissible and Bernoulli i.i.d. uniform

In a CIOQ switch with S>2,

MXM is stable under any traffic which satisfies the strong law of large numbers


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Future Questions We have seen that MSM is stable under the

auspices of batch scheduling

Perhaps we could incorporate this (well known) idea into a number of other algorithms to prove stability?

It would be nice to nail down the stability of MSM with uniform load in the absence of batch scheduling

Other open questions remain


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Backup


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Stability of MSM …2 Informal Arguments:

Similar to the CMSM proof, derive P{D < (1 + 1) Tk }

Use Chernoff bound, to derive P{mN > (1 - 2) Tk}

We can now write the probability of using less than

2[(1 + 1) Tk] – (1 - 2) Tk = (1 + 21 + 2)Tk time slots

Then rest of the proof is similar to CMSM

algorithm orals 2002 1 algorithm qualifying examination orals achieving 100% throughput in iq/cioq...

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